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• Level: GCSE
• Subject: Maths
• Word count: 3695

# In this investigation, I will attempt to find out some of the properties of a 2x2 square drawn within a 10x10 number grid. After this I hope to be able to investigate 9x9 and 8x8 grids, next I hope to move on to investigating rectangles.

Extracts from this document...

Introduction

Daniel Lovegrove                        10CW

Maths Coursework

MATHS

NumberGridCoursework

## Introduction

In this investigation, I will attempt to find out some of the properties of a 2x2 square drawn within a 10x10 number grid. After this I hope to be able to investigate 9x9 and 8x8 grids, next I hope to move on to investigating rectangles.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 96 97 98 100

such as

 35 36 45 46

when I find the product of the 2 sets of opposite corners I also find that the difference between these answers is ten:

2x2 squares

square                        products                 difference

 35 36 45 46

45 x 36 = 1620

35 x 46 = 1610

10

 57 58 67 68

57 x 68 = 3876

67 x 58 = 3886

10

 85 86 95 96

85 x 96 = 8160

95 x 86 = 8170

10

My findings are laid out here algebraically (s= starting number)

 35 36 45 46

The +10 in the lower left corner is there because every row has ten numbers and so if you look down the columns you will see that

here 35+10=40 and 36+10=46 and the

+1 is there obviously because every

number you count to the right is bigger

by 1 i.e. 35+1=36 I shall use the same method in all my working throughout this project

 S S+1 S+10 S+11

S(S+11) = S2 +11S

(S+10)(S+1) = S2 +10S +S + 10 = S2 + 11S + 10

The difference is ten

To find this I multiplied the corners of my square and found the difference between the products.

Next I investigated 3 x 3 squares within a 10 x 10 grid and found that the difference was 40

Square               products             difference

 1 2 3 11 12 13 21 22 23

1 x 23 = 23

21 x 3 = 63

40

 55 56 57 65 66 67 75 76 77

55 x 77 = 4235

75 x 57 = 4275

40

 44 45 46 54 55 56 64 65 66

44 x 66 = 2904

64 x 46 = 2944

40

Again my results are laid out algebraically (s = starting number)

 S S+1 S+2 S+10 S+11 S+12 S+20 S+21 S+22

S(S+22)

Middle

S(S+10) = S2+10S

(S+9)(S+1) = S2+10s+9

The difference is nine

My investigation continues with 3x3 squares

Square

Products

Difference

 29 30 31 38 39 40 47 48 49

29x49=1421

47x31=1457

36

 13 14 15 22 23 24 31 32 33

13x33=429

31x15=465

36

 48 49 50 57 58 59 66 67 68

48x68=3264

66x50=3300

36[2]

 S S+1 S+2 S+9 S+10 S+11 S+18 S+19 S+20

S(S+20) = S2+20S

(S+18)(S+2) = S2+20S+36

The difference is 36

Now I am investigating 4x4 squares

Square

Products

Difference

 30 31 32 33 39 40 41 42 48 49 50 51 57 58 59 60

30x60=1800

57x33=1881

81

 4 5 6 7 13 14 15 16 22 23 24 25 31 32 33 34

4x34=136

31x7=217

81

 22 23 24 25 31 32 33 34 40 41 42 43 49 50 51 52

22x52=1144

49x25=1225

81

 S S+1 S+2 S+3 S+9 S+10 S+11 S+12 S+18 S+19 S+20 S+21 S+27 S+28 S+29 S+30

S(S+30) = S2+30S

(S+27)(S+3) = S2+30S+81

the difference is 81

My second theory (see Above†) will be proved right or wrong by my investigations into 5x5 squares within a 9x9 grid.

Square

Products

Difference

 12 13 14 15 16 21 22 23 24 25 30 31 32 33 34 39 40 41 42 43 48 49 50 51 52

12x52=624

48x16=768

144

 41 42 43 44 45 50 51 52 53 54 59 60 61 62 63 68 69 70 71 72 77 78 79 80 81

41x81=3321

77x45=3465

144

 28 29 30 31 32 37 38 39 40 41 46 47 48 49 50 55 56 57 58 59 64 65 66 67 68

28x68=1904

64x32=2048

144

 S S+1 S+2 S+3 S+4 S+9 S+10 S+11 S+12 S+13 S+18 S+19 S+20 S+21 S+22 S+27 S+28 S+29 S+30 S+31 S+36 S+37 S+38 S+39 S+40

S(S+40) = S2+S40

(S+36)(S+4) = S2+S40+144

This is the correct difference

My second theory (see Above†) was correct

I am now going to produce a formula for squares in a 9x9 grid

 1 2 10 11

As before, I shall use my algebraic values to investigate 2x2, 3x3, 4x4 and 5x5 squares

W

 S S+(w-1) S+(9[w-1]) S+(9[w-1])+(w-1)

S(S+9W-9+W-1) = 1(11) =11

(S+9W-9)(S+W-1) = (10)(2)=20

the difference is 9, which is what it should be

 29 30 31 38 39 40 47 48 49
 S S+(W-1) S+9(W-1) S+9(W-1) +(W-1)

S(S+9[W-1]+[W-1]) = 29(49) = 1421

(S+9[W-1])(S+[W-1] = (47)(31) = 1457

the difference is 36, which is the correct difference

 22 23 24 25 31 32 33 34 40 41 42 43 49 50 51 52

Conclusion

(4X5)

Rectangle

Products

Difference

 51 52 53 54 61 62 63 64 71 72 73 74 81 82 83 84 91 92 93 94

51X94=4794

91X54=4914

120

 6 7 8 9 16 17 18 19 26 27 28 29 36 37 38 39 46 47 48 49

6X49=294

46X9=414

120

 1 2 3 4 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44

1X44=44

41X4=164

120

If I modify my formula for the difference of any size square in a 10X10 grid, I can create a formula for any size rectangle in a 10X10 grid.

• S=the starting number
• W=the width

Now I need one for the length I shall use O.

 24 25 34 35 44 45

To test my formula I shall use the rectangle

For this rectangle

O=3

W=2

S=24

 S S+(W-1) S+10(O-1) S+10(O-1)+(W-1)

S(S+10[O-1]+[W-1]) = 24(45) = 1080

(S+10[O-1])(S+[W-1]) = (44)(25) = 1100

the difference is 20 which is the correct number for this rectangle

Page  of

[1]* I now have a theory ; I think that the number in the difference column in a table for a 2x2 grid is the same as the size of the grid the 2x2 square is in, I shall investigate this further after I have completed my investigation of a 9x9 grid

[2]† I have another theory, I think that the numbers in the difference column for a square in a 9x9 grid are 90% of the difference of the corresponding size squares in a 10x10 grid just as the size of a 9x9 grid is 90% of the size of a 10x10

[3]‡ I am extending my second theory to say that the number in the difference column of a certain size square in an 8x8 grid is 80% of the same number for the same size square in a 10x10 grid and 70% for a 7x7 grid etc.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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